Found problems: 583
2018 AMC 10, 22
Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$?
$\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$
2013 IFYM, Sozopol, 7
Let $T$ be a set of natural numbers, each of which is greater than 1. A subset $S$ of $T$ is called “good”, if for each $t\in T$ there exists $s\in S$, for which $gcd(t,s)>1$. Prove that the number of "good" subsets of $T$ is odd.
1981 USAMO, 1
The measure of a given angle is $\frac{180^{\circ}}{n}$ where $n$ is a positive integer not divisible by $3$. Prove that the angle can be trisected by Euclidean means (straightedge and compasses).
2008 Bundeswettbewerb Mathematik, 2
Let the positive integers $ a,b,c$ chosen such that the quotients $ \frac{bc}{b\plus{}c},$ $ \frac{ca}{c\plus{}a}$ and $ \frac{ab}{a\plus{}b}$ are integers. Prove that $ a,b,c$ have a common divisor greater than 1.
2016 Greece Team Selection Test, 4
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
1949 Miklós Schweitzer, 6
Let $ n$ and $ k$ be positive integers, $ n\geq k$. Prove that the greatest common divisor of the numbers $ \binom{n}{k},\binom{n\plus{}1}{k},\ldots,\binom{n\plus{}k}{k}$ is $ 1$.
2009 China Second Round Olympiad, 3
Let $k,l$ be two given integers. Prove that there exist infinite many integers $m\ge k$ such that $\gcd\left(\binom{m}{k},l\right)=1$.
2023 AMC 10, 18
Suppose $a$, $b$, and $c$ are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true?
I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$.
II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.
III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$.
$\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$
2006 AMC 12/AHSME, 25
A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n \plus{} 2} \equal{} |a_{n \plus{} 1} \minus{} a_n|$ for $ n\ge 1$. If $ a_1 \equal{} 999, a_2 < 999,$ and $ a_{2006} \equal{} 1$, how many different values of $ a_2$ are possible?
$ \textbf{(A) } 165 \qquad \textbf{(B) } 324 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 499 \qquad \textbf{(E) } 660$
2016 Macedonia JBMO TST, 5
Solve the following equation in the set of positive integers
$x + y^2 + (GCD(x, y))^2 = xy \cdot GCD(x, y)$.
2004 Thailand Mathematical Olympiad, 14
Compute gcd$(5^{2547} - 1, 5^{2004} - 1)$.
2014 Contests, 3
For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers.
(a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$.
(b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.
2017 Brazil National Olympiad, 2.
[b]2.[/b] Let $n \geq 3$ be an integer. Prove that for all integers $k$, with $1 \leq k \leq \binom{n}{2}$, there exists a set $A$ with $n$ distinct positive integer elements such that the set $B = \{\gcd(x, y): x, y \in A, x \neq y \}$ (gotten from the greatest common divisor of all pairs of distinct elements from $A$) contains exactly $k$ distinct elements.
2002 Turkey Junior National Olympiad, 3
Find all ordered positive integer pairs of $(m,n)$ such that $2^n-1$ divides $2^m+1$.
2020 Mexico National Olympiad, 1
A set of five different positive integers is called [i]virtual[/i] if the greatest common divisor of any three of its elements is greater than $1$, but the greatest common divisor of any four of its elements is equal to $1$. Prove that, in any virtual set, the product of its elements has at least $2020$ distinct positive divisors.
[i]Proposed by Víctor Almendra[/i]
1997 Putnam, 3
For each positive integer $n$ write the sum $\sum_{i=}^{n}\frac{1}{i}=\frac{p_n}{q_n}$ with $\text{gcd}(p_n,q_n)=1$. Find all such $n$ such that $5\nmid q_n$.
2018-IMOC, N2
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$
for all $x,y\in\mathbb N$.
Oliforum Contest I 2008, 2
Find all non-negative integers $ x,y,z$ such that $ 5^x \plus{} 7^y \equal{} 2^z$.
:lol:
([i]Daniel Kohen, University of Buenos Aires - Buenos Aires,Argentina[/i])
2022 Germany Team Selection Test, 1
Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
2021 IMO Shortlist, C1
Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
2014 Canadian Mathematical Olympiad Qualification, 1
Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$. If $h(x, y)$ is a two-variable polynomial in $x$ and $y$, prove that it must be constant.
2006 Pre-Preparation Course Examination, 3
a) If $K$ is a finite extension of the field $F$ and $K=F(\alpha,\beta)$ show that $[K: F]\leq [F(\alpha): F][F(\beta): F]$
b) If $gcd([F(\alpha): F],[F(\beta): F])=1$ then does the above inequality always become equality?
c) By giving an example show that if $gcd([F(\alpha): F],[F(\beta): F])\neq 1$ then equality might happen.
1982 IMO Longlists, 7
Find all solutions $(x, y) \in \mathbb Z^2$ of the equation
\[x^3 - y^3 = 2xy + 8.\]
1991 Romania Team Selection Test, 8
Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$.
If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.
2008 Indonesia TST, 3
Let $n$ be an arbitrary positive integer.
(a) For every positive integers $a$ and $b$, show that $gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1$.
(b) Show that there exist infinitely many composite pairs ($a, b)$, such that each of them is not a multiply of the other number and equality holds in (a).